Self-similar measures are “locally nice” by nature of their construction (indeed, they have equal box and Hausdorff dimensions [1]}), so one might be more optimistic for nice multifractal properties.
For example, self-similar measures are exact-dimensional [2]}, which means that there is precisely one value \(\alpha \)  for which the level set \(\lbrace x\in \mu :\dim _{}(\mu ,x)=\alpha \rbrace \)  has full \(\mu \) -measure.
If there is an open set \(U\)  satisfying \(\bigcup _{i\in \mathcal {I}}S_i(U)\subseteq U\)  where the union is disjoint, we say that \(\mu \)  satisfies the open set condition [3]}.
For such measures, the \(L^q\) -spectrum is the unique smooth function satisfying \(\sum _{i\in \mathcal {I}}p_i^qr_i^{-\tau _\mu (q)}=1\) , and the multifractal formalism holds [4]}, [5]}.
